3.29.16 \(\int \frac {b^2+a^2 x^2}{(-b^2+a^2 x^2)^3 \sqrt [3]{-b x^2+a x^3}} \, dx\)

Optimal. Leaf size=278 \[ -\frac {59 \log \left (2 a^{2/3} x^2+2^{2/3} \sqrt [3]{a} x \sqrt [3]{a x^3-b x^2}+\sqrt [3]{2} \left (a x^3-b x^2\right )^{2/3}\right )}{288 \sqrt [3]{2} \sqrt [3]{a} b^4}-\frac {\left (a x^3-b x^2\right )^{2/3} \left (-625 a^4 x^4-67 a^3 b x^3+1503 a^2 b^2 x^2+91 a b^3 x-1190 b^4\right )}{672 b^4 x (b-a x)^3 (a x+b)^2}+\frac {59 \log \left (2^{2/3} \sqrt [3]{a x^3-b x^2}-2 \sqrt [3]{a} x\right )}{144 \sqrt [3]{2} \sqrt [3]{a} b^4}-\frac {59 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{a} x}{2^{2/3} \sqrt [3]{a x^3-b x^2}+\sqrt [3]{a} x}\right )}{48 \sqrt [3]{2} \sqrt {3} \sqrt [3]{a} b^4} \]

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Rubi [C]  time = 147.51, antiderivative size = 1306, normalized size of antiderivative = 4.70, number of steps used = 10, number of rules used = 6, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {2056, 6733, 6742, 1404, 430, 429}

result too large to display

Warning: Unable to verify antiderivative.

[In]

Int[(b^2 + a^2*x^2)/((-b^2 + a^2*x^2)^3*(-(b*x^2) + a*x^3)^(1/3)),x]

[Out]

(9*x*Gamma[4/3]*(1820*b^4*Hypergeometric2F1[1, 2, 13/3, (2*a*x)/(b + a*x)] - 2275*a*b^3*x*Hypergeometric2F1[1,
2, 13/3, (2*a*x)/(b + a*x)] - 585*a^2*b^2*x^2*Hypergeometric2F1[1, 2, 13/3, (2*a*x)/(b + a*x)] + 2457*a^3*b*x
^3*Hypergeometric2F1[1, 2, 13/3, (2*a*x)/(b + a*x)] - 1053*a^4*x^4*Hypergeometric2F1[1, 2, 13/3, (2*a*x)/(b +
a*x)] + 2484*a*b^3*x*Hypergeometric2F1[2, 3, 16/3, (2*a*x)/(b + a*x)] - 6534*a^2*b^2*x^2*Hypergeometric2F1[2,
3, 16/3, (2*a*x)/(b + a*x)] + 5832*a^3*b*x^3*Hypergeometric2F1[2, 3, 16/3, (2*a*x)/(b + a*x)] - 1782*a^4*x^4*H
ypergeometric2F1[2, 3, 16/3, (2*a*x)/(b + a*x)] + 162*a*x*(7*b - 6*a*x)*(b - a*x)^2*HypergeometricPFQ[{2, 2, 3
}, {1, 16/3}, (2*a*x)/(b + a*x)] + 162*a*(b*x^(1/3) - a*x^(4/3))^3*HypergeometricPFQ[{2, 2, 2, 3}, {1, 1, 16/3
}, (2*a*x)/(b + a*x)]))/(1820*b^4*(b - a*x)*(b + a*x)^3*(-(b*x^2) + a*x^3)^(1/3)*Gamma[1/3]) - (9*x*Gamma[4/3]
*(553280*b^6*Hypergeometric2F1[1, 3, 19/3, (2*a*x)/(b + a*x)] - 1521520*a*b^5*x*Hypergeometric2F1[1, 3, 19/3,
(2*a*x)/(b + a*x)] + 1482000*a^2*b^4*x^2*Hypergeometric2F1[1, 3, 19/3, (2*a*x)/(b + a*x)] + 355680*a^3*b^3*x^3
*Hypergeometric2F1[1, 3, 19/3, (2*a*x)/(b + a*x)] - 1723680*a^4*b^2*x^4*Hypergeometric2F1[1, 3, 19/3, (2*a*x)/
(b + a*x)] + 1200420*a^5*b*x^5*Hypergeometric2F1[1, 3, 19/3, (2*a*x)/(b + a*x)] - 277020*a^6*x^6*Hypergeometri
c2F1[1, 3, 19/3, (2*a*x)/(b + a*x)] + 994248*a*b^5*x*Hypergeometric2F1[2, 4, 22/3, (2*a*x)/(b + a*x)] - 421983
0*a^2*b^4*x^2*Hypergeometric2F1[2, 4, 22/3, (2*a*x)/(b + a*x)] + 7474680*a^3*b^3*x^3*Hypergeometric2F1[2, 4, 2
2/3, (2*a*x)/(b + a*x)] - 6828300*a^4*b^2*x^4*Hypergeometric2F1[2, 4, 22/3, (2*a*x)/(b + a*x)] + 3178440*a^5*b
*x^5*Hypergeometric2F1[2, 4, 22/3, (2*a*x)/(b + a*x)] - 599238*a^6*x^6*Hypergeometric2F1[2, 4, 22/3, (2*a*x)/(
b + a*x)] + 405*a*x*(b - a*x)^2*(1730*b^3 - 4419*a*b^2*x + 3960*a^2*b*x^2 - 1215*a^3*x^3)*HypergeometricPFQ[{2
, 2, 4}, {1, 22/3}, (2*a*x)/(b + a*x)] + 1215*a*(b*x^(1/3) - a*x^(4/3))^3*(191*b^2 - 336*a*b*x + 153*a^2*x^2)*
HypergeometricPFQ[{2, 2, 2, 4}, {1, 1, 22/3}, (2*a*x)/(b + a*x)] + 36450*a*b^5*x*HypergeometricPFQ[{2, 2, 2, 2
, 4}, {1, 1, 1, 22/3}, (2*a*x)/(b + a*x)] - 178605*a^2*b^4*x^2*HypergeometricPFQ[{2, 2, 2, 2, 4}, {1, 1, 1, 22
/3}, (2*a*x)/(b + a*x)] + 349920*a^3*b^3*x^3*HypergeometricPFQ[{2, 2, 2, 2, 4}, {1, 1, 1, 22/3}, (2*a*x)/(b +
a*x)] - 342630*a^4*b^2*x^4*HypergeometricPFQ[{2, 2, 2, 2, 4}, {1, 1, 1, 22/3}, (2*a*x)/(b + a*x)] + 167670*a^5
*b*x^5*HypergeometricPFQ[{2, 2, 2, 2, 4}, {1, 1, 1, 22/3}, (2*a*x)/(b + a*x)] - 32805*a^6*x^6*HypergeometricPF
Q[{2, 2, 2, 2, 4}, {1, 1, 1, 22/3}, (2*a*x)/(b + a*x)] + 2187*a*b^5*x*HypergeometricPFQ[{2, 2, 2, 2, 2, 4}, {1
, 1, 1, 1, 22/3}, (2*a*x)/(b + a*x)] - 10935*a^2*b^4*x^2*HypergeometricPFQ[{2, 2, 2, 2, 2, 4}, {1, 1, 1, 1, 22
/3}, (2*a*x)/(b + a*x)] + 21870*a^3*b^3*x^3*HypergeometricPFQ[{2, 2, 2, 2, 2, 4}, {1, 1, 1, 1, 22/3}, (2*a*x)/
(b + a*x)] - 21870*a^4*b^2*x^4*HypergeometricPFQ[{2, 2, 2, 2, 2, 4}, {1, 1, 1, 1, 22/3}, (2*a*x)/(b + a*x)] +
10935*a^5*b*x^5*HypergeometricPFQ[{2, 2, 2, 2, 2, 4}, {1, 1, 1, 1, 22/3}, (2*a*x)/(b + a*x)] - 2187*a^6*x^6*Hy
pergeometricPFQ[{2, 2, 2, 2, 2, 4}, {1, 1, 1, 1, 22/3}, (2*a*x)/(b + a*x)]))/(276640*b^4*(b - a*x)^2*(b + a*x)
^4*(-(b*x^2) + a*x^3)^(1/3)*Gamma[1/3])

Rule 429

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 430

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^F
racPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n,
p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 1404

Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((a_) + (c_.)*(x_)^(n2_))^(p_.), x_Symbol] :> Int[(d + e*x^n)^(p + q)*(a/d
+ (c*x^n)/e)^p, x] /; FreeQ[{a, c, d, e, n, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && IntegerQ[p]

Rule 2056

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6733

Int[(u_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k, Subst[Int[x^(k*(m + 1) - 1)*(u /. x -> x^k
), x], x, x^(1/k)], x]] /; FractionQ[m]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {b^2+a^2 x^2}{\left (-b^2+a^2 x^2\right )^3 \sqrt [3]{-b x^2+a x^3}} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {b^2+a^2 x^2}{x^{2/3} \sqrt [3]{-b+a x} \left (-b^2+a^2 x^2\right )^3} \, dx}{\sqrt [3]{-b x^2+a x^3}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {b^2+a^2 x^6}{\sqrt [3]{-b+a x^3} \left (-b^2+a^2 x^6\right )^3} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-b x^2+a x^3}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-b+a x}\right ) \operatorname {Subst}\left (\int \left (\frac {2 b^2}{\sqrt [3]{-b+a x^3} \left (-b^2+a^2 x^6\right )^3}+\frac {1}{\sqrt [3]{-b+a x^3} \left (-b^2+a^2 x^6\right )^2}\right ) \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-b x^2+a x^3}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-b+a x^3} \left (-b^2+a^2 x^6\right )^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-b x^2+a x^3}}+\frac {\left (6 b^2 x^{2/3} \sqrt [3]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-b+a x^3} \left (-b^2+a^2 x^6\right )^3} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-b x^2+a x^3}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-b+a x^3\right )^{7/3} \left (b+a x^3\right )^2} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-b x^2+a x^3}}+\frac {\left (6 b^2 x^{2/3} \sqrt [3]{-b+a x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-b+a x^3\right )^{10/3} \left (b+a x^3\right )^3} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-b x^2+a x^3}}\\ &=\frac {\left (3 x^{2/3} \sqrt [3]{1-\frac {a x}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b+a x^3\right )^2 \left (1-\frac {a x^3}{b}\right )^{7/3}} \, dx,x,\sqrt [3]{x}\right )}{b^2 \sqrt [3]{-b x^2+a x^3}}-\frac {\left (6 x^{2/3} \sqrt [3]{1-\frac {a x}{b}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (b+a x^3\right )^3 \left (1-\frac {a x^3}{b}\right )^{10/3}} \, dx,x,\sqrt [3]{x}\right )}{b \sqrt [3]{-b x^2+a x^3}}\\ &=\frac {9 x \Gamma \left (\frac {4}{3}\right ) \left (1820 b^4 \, _2F_1\left (1,2;\frac {13}{3};\frac {2 a x}{b+a x}\right )-2275 a b^3 x \, _2F_1\left (1,2;\frac {13}{3};\frac {2 a x}{b+a x}\right )-585 a^2 b^2 x^2 \, _2F_1\left (1,2;\frac {13}{3};\frac {2 a x}{b+a x}\right )+2457 a^3 b x^3 \, _2F_1\left (1,2;\frac {13}{3};\frac {2 a x}{b+a x}\right )-1053 a^4 x^4 \, _2F_1\left (1,2;\frac {13}{3};\frac {2 a x}{b+a x}\right )+2484 a b^3 x \, _2F_1\left (2,3;\frac {16}{3};\frac {2 a x}{b+a x}\right )-6534 a^2 b^2 x^2 \, _2F_1\left (2,3;\frac {16}{3};\frac {2 a x}{b+a x}\right )+5832 a^3 b x^3 \, _2F_1\left (2,3;\frac {16}{3};\frac {2 a x}{b+a x}\right )-1782 a^4 x^4 \, _2F_1\left (2,3;\frac {16}{3};\frac {2 a x}{b+a x}\right )+162 a x (7 b-6 a x) (b-a x)^2 \, _3F_2\left (2,2,3;1,\frac {16}{3};\frac {2 a x}{b+a x}\right )+162 a \left (b \sqrt [3]{x}-a x^{4/3}\right )^3 \, _4F_3\left (2,2,2,3;1,1,\frac {16}{3};\frac {2 a x}{b+a x}\right )\right )}{1820 b^4 (b-a x) (b+a x)^3 \sqrt [3]{-b x^2+a x^3} \Gamma \left (\frac {1}{3}\right )}-\frac {9 x \Gamma \left (\frac {4}{3}\right ) \left (553280 b^6 \, _2F_1\left (1,3;\frac {19}{3};\frac {2 a x}{b+a x}\right )-1521520 a b^5 x \, _2F_1\left (1,3;\frac {19}{3};\frac {2 a x}{b+a x}\right )+1482000 a^2 b^4 x^2 \, _2F_1\left (1,3;\frac {19}{3};\frac {2 a x}{b+a x}\right )+355680 a^3 b^3 x^3 \, _2F_1\left (1,3;\frac {19}{3};\frac {2 a x}{b+a x}\right )-1723680 a^4 b^2 x^4 \, _2F_1\left (1,3;\frac {19}{3};\frac {2 a x}{b+a x}\right )+1200420 a^5 b x^5 \, _2F_1\left (1,3;\frac {19}{3};\frac {2 a x}{b+a x}\right )-277020 a^6 x^6 \, _2F_1\left (1,3;\frac {19}{3};\frac {2 a x}{b+a x}\right )+994248 a b^5 x \, _2F_1\left (2,4;\frac {22}{3};\frac {2 a x}{b+a x}\right )-4219830 a^2 b^4 x^2 \, _2F_1\left (2,4;\frac {22}{3};\frac {2 a x}{b+a x}\right )+7474680 a^3 b^3 x^3 \, _2F_1\left (2,4;\frac {22}{3};\frac {2 a x}{b+a x}\right )-6828300 a^4 b^2 x^4 \, _2F_1\left (2,4;\frac {22}{3};\frac {2 a x}{b+a x}\right )+3178440 a^5 b x^5 \, _2F_1\left (2,4;\frac {22}{3};\frac {2 a x}{b+a x}\right )-599238 a^6 x^6 \, _2F_1\left (2,4;\frac {22}{3};\frac {2 a x}{b+a x}\right )+405 a x (b-a x)^2 \left (1730 b^3-4419 a b^2 x+3960 a^2 b x^2-1215 a^3 x^3\right ) \, _3F_2\left (2,2,4;1,\frac {22}{3};\frac {2 a x}{b+a x}\right )+1215 a \left (b \sqrt [3]{x}-a x^{4/3}\right )^3 \left (191 b^2-336 a b x+153 a^2 x^2\right ) \, _4F_3\left (2,2,2,4;1,1,\frac {22}{3};\frac {2 a x}{b+a x}\right )+36450 a b^5 x \, _5F_4\left (2,2,2,2,4;1,1,1,\frac {22}{3};\frac {2 a x}{b+a x}\right )-178605 a^2 b^4 x^2 \, _5F_4\left (2,2,2,2,4;1,1,1,\frac {22}{3};\frac {2 a x}{b+a x}\right )+349920 a^3 b^3 x^3 \, _5F_4\left (2,2,2,2,4;1,1,1,\frac {22}{3};\frac {2 a x}{b+a x}\right )-342630 a^4 b^2 x^4 \, _5F_4\left (2,2,2,2,4;1,1,1,\frac {22}{3};\frac {2 a x}{b+a x}\right )+167670 a^5 b x^5 \, _5F_4\left (2,2,2,2,4;1,1,1,\frac {22}{3};\frac {2 a x}{b+a x}\right )-32805 a^6 x^6 \, _5F_4\left (2,2,2,2,4;1,1,1,\frac {22}{3};\frac {2 a x}{b+a x}\right )+2187 a b^5 x \, _6F_5\left (2,2,2,2,2,4;1,1,1,1,\frac {22}{3};\frac {2 a x}{b+a x}\right )-10935 a^2 b^4 x^2 \, _6F_5\left (2,2,2,2,2,4;1,1,1,1,\frac {22}{3};\frac {2 a x}{b+a x}\right )+21870 a^3 b^3 x^3 \, _6F_5\left (2,2,2,2,2,4;1,1,1,1,\frac {22}{3};\frac {2 a x}{b+a x}\right )-21870 a^4 b^2 x^4 \, _6F_5\left (2,2,2,2,2,4;1,1,1,1,\frac {22}{3};\frac {2 a x}{b+a x}\right )+10935 a^5 b x^5 \, _6F_5\left (2,2,2,2,2,4;1,1,1,1,\frac {22}{3};\frac {2 a x}{b+a x}\right )-2187 a^6 x^6 \, _6F_5\left (2,2,2,2,2,4;1,1,1,1,\frac {22}{3};\frac {2 a x}{b+a x}\right )\right )}{276640 b^4 (b-a x)^2 (b+a x)^4 \sqrt [3]{-b x^2+a x^3} \Gamma \left (\frac {1}{3}\right )}\\ \end {align*}

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Mathematica [C]  time = 0.88, size = 156, normalized size = 0.56 \begin {gather*} \frac {x \sqrt [3]{\frac {a x}{b}+1} \left (-625 a^4 x^4-67 a^3 b x^3+1503 a^2 b^2 x^2+91 a b^3 x-1190 b^4\right )-826 x \sqrt [3]{1-\frac {a x}{b}} \left (b^2-a^2 x^2\right )^2 \, _2F_1\left (\frac {1}{3},\frac {1}{3};\frac {4}{3};\frac {2 a x}{b+a x}\right )}{672 b^4 (b-a x)^2 \sqrt [3]{x^2 (a x-b)} (a x+b)^2 \sqrt [3]{\frac {a x}{b}+1}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(b^2 + a^2*x^2)/((-b^2 + a^2*x^2)^3*(-(b*x^2) + a*x^3)^(1/3)),x]

[Out]

(x*(1 + (a*x)/b)^(1/3)*(-1190*b^4 + 91*a*b^3*x + 1503*a^2*b^2*x^2 - 67*a^3*b*x^3 - 625*a^4*x^4) - 826*x*(1 - (
a*x)/b)^(1/3)*(b^2 - a^2*x^2)^2*Hypergeometric2F1[1/3, 1/3, 4/3, (2*a*x)/(b + a*x)])/(672*b^4*(b - a*x)^2*(x^2
*(-b + a*x))^(1/3)*(b + a*x)^2*(1 + (a*x)/b)^(1/3))

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IntegrateAlgebraic [A]  time = 1.10, size = 278, normalized size = 1.00 \begin {gather*} -\frac {\left (-b x^2+a x^3\right )^{2/3} \left (-1190 b^4+91 a b^3 x+1503 a^2 b^2 x^2-67 a^3 b x^3-625 a^4 x^4\right )}{672 b^4 x (b-a x)^3 (b+a x)^2}-\frac {59 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{a} x}{\sqrt [3]{a} x+2^{2/3} \sqrt [3]{-b x^2+a x^3}}\right )}{48 \sqrt [3]{2} \sqrt {3} \sqrt [3]{a} b^4}+\frac {59 \log \left (-2 \sqrt [3]{a} x+2^{2/3} \sqrt [3]{-b x^2+a x^3}\right )}{144 \sqrt [3]{2} \sqrt [3]{a} b^4}-\frac {59 \log \left (2 a^{2/3} x^2+2^{2/3} \sqrt [3]{a} x \sqrt [3]{-b x^2+a x^3}+\sqrt [3]{2} \left (-b x^2+a x^3\right )^{2/3}\right )}{288 \sqrt [3]{2} \sqrt [3]{a} b^4} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(b^2 + a^2*x^2)/((-b^2 + a^2*x^2)^3*(-(b*x^2) + a*x^3)^(1/3)),x]

[Out]

-1/672*((-(b*x^2) + a*x^3)^(2/3)*(-1190*b^4 + 91*a*b^3*x + 1503*a^2*b^2*x^2 - 67*a^3*b*x^3 - 625*a^4*x^4))/(b^
4*x*(b - a*x)^3*(b + a*x)^2) - (59*ArcTan[(Sqrt[3]*a^(1/3)*x)/(a^(1/3)*x + 2^(2/3)*(-(b*x^2) + a*x^3)^(1/3))])
/(48*2^(1/3)*Sqrt[3]*a^(1/3)*b^4) + (59*Log[-2*a^(1/3)*x + 2^(2/3)*(-(b*x^2) + a*x^3)^(1/3)])/(144*2^(1/3)*a^(
1/3)*b^4) - (59*Log[2*a^(2/3)*x^2 + 2^(2/3)*a^(1/3)*x*(-(b*x^2) + a*x^3)^(1/3) + 2^(1/3)*(-(b*x^2) + a*x^3)^(2
/3)])/(288*2^(1/3)*a^(1/3)*b^4)

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fricas [A]  time = 0.52, size = 959, normalized size = 3.45 \begin {gather*} \left [\frac {826 \cdot 2^{\frac {2}{3}} {\left (a^{5} x^{6} - a^{4} b x^{5} - 2 \, a^{3} b^{2} x^{4} + 2 \, a^{2} b^{3} x^{3} + a b^{4} x^{2} - b^{5} x\right )} a^{\frac {2}{3}} \log \left (-\frac {2^{\frac {1}{3}} a^{\frac {1}{3}} x - {\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - 413 \cdot 2^{\frac {2}{3}} {\left (a^{5} x^{6} - a^{4} b x^{5} - 2 \, a^{3} b^{2} x^{4} + 2 \, a^{2} b^{3} x^{3} + a b^{4} x^{2} - b^{5} x\right )} a^{\frac {2}{3}} \log \left (\frac {2^{\frac {2}{3}} a^{\frac {2}{3}} x^{2} + 2^{\frac {1}{3}} {\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}} a^{\frac {1}{3}} x + {\left (a x^{3} - b x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) + 2478 \, \sqrt {\frac {1}{6}} {\left (a^{6} x^{6} - a^{5} b x^{5} - 2 \, a^{4} b^{2} x^{4} + 2 \, a^{3} b^{3} x^{3} + a^{2} b^{4} x^{2} - a b^{5} x\right )} \sqrt {-\frac {2^{\frac {1}{3}}}{a^{\frac {2}{3}}}} \log \left (-\frac {4 \, a x^{2} - 3 \cdot 2^{\frac {2}{3}} {\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}} a^{\frac {2}{3}} x - 2 \, b x - 6 \, \sqrt {\frac {1}{6}} {\left (2^{\frac {1}{3}} a^{\frac {4}{3}} x^{2} + {\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}} a x - 2^{\frac {2}{3}} {\left (a x^{3} - b x^{2}\right )}^{\frac {2}{3}} a^{\frac {2}{3}}\right )} \sqrt {-\frac {2^{\frac {1}{3}}}{a^{\frac {2}{3}}}}}{a x^{2} + b x}\right ) - 6 \, {\left (625 \, a^{5} x^{4} + 67 \, a^{4} b x^{3} - 1503 \, a^{3} b^{2} x^{2} - 91 \, a^{2} b^{3} x + 1190 \, a b^{4}\right )} {\left (a x^{3} - b x^{2}\right )}^{\frac {2}{3}}}{4032 \, {\left (a^{6} b^{4} x^{6} - a^{5} b^{5} x^{5} - 2 \, a^{4} b^{6} x^{4} + 2 \, a^{3} b^{7} x^{3} + a^{2} b^{8} x^{2} - a b^{9} x\right )}}, \frac {826 \cdot 2^{\frac {2}{3}} {\left (a^{5} x^{6} - a^{4} b x^{5} - 2 \, a^{3} b^{2} x^{4} + 2 \, a^{2} b^{3} x^{3} + a b^{4} x^{2} - b^{5} x\right )} a^{\frac {2}{3}} \log \left (-\frac {2^{\frac {1}{3}} a^{\frac {1}{3}} x - {\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - 413 \cdot 2^{\frac {2}{3}} {\left (a^{5} x^{6} - a^{4} b x^{5} - 2 \, a^{3} b^{2} x^{4} + 2 \, a^{2} b^{3} x^{3} + a b^{4} x^{2} - b^{5} x\right )} a^{\frac {2}{3}} \log \left (\frac {2^{\frac {2}{3}} a^{\frac {2}{3}} x^{2} + 2^{\frac {1}{3}} {\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}} a^{\frac {1}{3}} x + {\left (a x^{3} - b x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) + 4956 \, \sqrt {\frac {1}{6}} {\left (a^{6} x^{6} - a^{5} b x^{5} - 2 \, a^{4} b^{2} x^{4} + 2 \, a^{3} b^{3} x^{3} + a^{2} b^{4} x^{2} - a b^{5} x\right )} \sqrt {\frac {2^{\frac {1}{3}}}{a^{\frac {2}{3}}}} \arctan \left (\frac {\sqrt {\frac {1}{6}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} x + 2 \, {\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}}\right )} \sqrt {\frac {2^{\frac {1}{3}}}{a^{\frac {2}{3}}}}}{x}\right ) - 6 \, {\left (625 \, a^{5} x^{4} + 67 \, a^{4} b x^{3} - 1503 \, a^{3} b^{2} x^{2} - 91 \, a^{2} b^{3} x + 1190 \, a b^{4}\right )} {\left (a x^{3} - b x^{2}\right )}^{\frac {2}{3}}}{4032 \, {\left (a^{6} b^{4} x^{6} - a^{5} b^{5} x^{5} - 2 \, a^{4} b^{6} x^{4} + 2 \, a^{3} b^{7} x^{3} + a^{2} b^{8} x^{2} - a b^{9} x\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*x^2+b^2)/(a^2*x^2-b^2)^3/(a*x^3-b*x^2)^(1/3),x, algorithm="fricas")

[Out]

[1/4032*(826*2^(2/3)*(a^5*x^6 - a^4*b*x^5 - 2*a^3*b^2*x^4 + 2*a^2*b^3*x^3 + a*b^4*x^2 - b^5*x)*a^(2/3)*log(-(2
^(1/3)*a^(1/3)*x - (a*x^3 - b*x^2)^(1/3))/x) - 413*2^(2/3)*(a^5*x^6 - a^4*b*x^5 - 2*a^3*b^2*x^4 + 2*a^2*b^3*x^
3 + a*b^4*x^2 - b^5*x)*a^(2/3)*log((2^(2/3)*a^(2/3)*x^2 + 2^(1/3)*(a*x^3 - b*x^2)^(1/3)*a^(1/3)*x + (a*x^3 - b
*x^2)^(2/3))/x^2) + 2478*sqrt(1/6)*(a^6*x^6 - a^5*b*x^5 - 2*a^4*b^2*x^4 + 2*a^3*b^3*x^3 + a^2*b^4*x^2 - a*b^5*
x)*sqrt(-2^(1/3)/a^(2/3))*log(-(4*a*x^2 - 3*2^(2/3)*(a*x^3 - b*x^2)^(1/3)*a^(2/3)*x - 2*b*x - 6*sqrt(1/6)*(2^(
1/3)*a^(4/3)*x^2 + (a*x^3 - b*x^2)^(1/3)*a*x - 2^(2/3)*(a*x^3 - b*x^2)^(2/3)*a^(2/3))*sqrt(-2^(1/3)/a^(2/3)))/
(a*x^2 + b*x)) - 6*(625*a^5*x^4 + 67*a^4*b*x^3 - 1503*a^3*b^2*x^2 - 91*a^2*b^3*x + 1190*a*b^4)*(a*x^3 - b*x^2)
^(2/3))/(a^6*b^4*x^6 - a^5*b^5*x^5 - 2*a^4*b^6*x^4 + 2*a^3*b^7*x^3 + a^2*b^8*x^2 - a*b^9*x), 1/4032*(826*2^(2/
3)*(a^5*x^6 - a^4*b*x^5 - 2*a^3*b^2*x^4 + 2*a^2*b^3*x^3 + a*b^4*x^2 - b^5*x)*a^(2/3)*log(-(2^(1/3)*a^(1/3)*x -
(a*x^3 - b*x^2)^(1/3))/x) - 413*2^(2/3)*(a^5*x^6 - a^4*b*x^5 - 2*a^3*b^2*x^4 + 2*a^2*b^3*x^3 + a*b^4*x^2 - b^
5*x)*a^(2/3)*log((2^(2/3)*a^(2/3)*x^2 + 2^(1/3)*(a*x^3 - b*x^2)^(1/3)*a^(1/3)*x + (a*x^3 - b*x^2)^(2/3))/x^2)
+ 4956*sqrt(1/6)*(a^6*x^6 - a^5*b*x^5 - 2*a^4*b^2*x^4 + 2*a^3*b^3*x^3 + a^2*b^4*x^2 - a*b^5*x)*sqrt(2^(1/3)/a^
(2/3))*arctan(sqrt(1/6)*(2^(1/3)*a^(1/3)*x + 2*(a*x^3 - b*x^2)^(1/3))*sqrt(2^(1/3)/a^(2/3))/x) - 6*(625*a^5*x^
4 + 67*a^4*b*x^3 - 1503*a^3*b^2*x^2 - 91*a^2*b^3*x + 1190*a*b^4)*(a*x^3 - b*x^2)^(2/3))/(a^6*b^4*x^6 - a^5*b^5
*x^5 - 2*a^4*b^6*x^4 + 2*a^3*b^7*x^3 + a^2*b^8*x^2 - a*b^9*x)]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*x^2+b^2)/(a^2*x^2-b^2)^3/(a*x^3-b*x^2)^(1/3),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Algebraic extensions not allowed in a ro
otofAlgebraic extensions not allowed in a rootofAlgebraic extensions not allowed in a rootofAlgebraic extensio
ns not allowed in a rootofAlgebraic extensions not allowed in a rootofAlgebraic extensions not allowed in a ro
otofAlgebraic extensions not allowed in a rootofAlgebraic extensions not allowed in a rootofAlgebraic extensio
ns not allowed in a rootofAlgebraic extensions not allowed in a rootofAlgebraic extensions not allowed in a ro
otofAlgebraic extensions not allowed in a rootofAlgebraic extensions not allowed in a rootofAlgebraic extensio
ns not allowed in a rootofAlgebraic extensions not allowed in a rootofAlgebraic extensions not allowed in a ro
otofAlgebraic extensions not allowed in a rootofAlgebraic extensions not allowed in a rootofAlgebraic extensio
ns not allowed in a rootofAlgebraic extensions not allowed in a rootofAlgebraic extensions not allowed in a ro
otofAlgebraic extensions not allowed in a rootofAlgebraic extensions not allowed in a rootofAlgebraic extensio
ns not allowed in a rootofAlgebraic extensions not allowed in a rootofAlgebraic extensions not allowed in a ro
otofAlgebraic extensions not allowed in a rootofAlgebraic extensions not allowed in a rootofAlgebraic extensio
ns not allowed in a rootofAlgebraic extensions not allowed in a rootofAlgebraic extensions not allowed in a ro
otofAlgebraic extensions not allowed in a rootofAlgebraic extensions not allowed in a rootofAlgebraic extensio
ns not allowed in a rootofAlgebraic extensions not allowed in a rootofAlgebraic extensions not allowed in a ro
otofAlgebraic extensions not allowed in a rootofAlgebraic extensions not allowed in a rootofAlgebraic extensio
ns not allowed in a rootofAlgebraic extensions not allowed in a rootofAlgebraic extensions not allowed in a ro
otofAlgebraic extensions not allowed in a rootofAlgebraic extensions not allowed in a rootofAlgebraic extensio
ns not allowed in a rootofAlgebraic extensions not allowed in a rootofAlgebraic extensions not allowed in a ro
otofAlgebraic extensions not allowed in a rootofAlgebraic extensions not allowed in a rootof-(13*((a-b/x)^(1/3
))^2*(a-b/x)-23*((a-b/x)^(1/3))^2*a)/48/b^4/(a-b/x-2*a)^2+(-336*(a-b/x)^2+105*(a-b/x)*a-24*a^2)/224/b^4/(a-b/x
)^(1/3)/(a-b/x)^2-59/288/(2*a)^(1/3)/b^4*ln(((a-b/x)^(1/3))^2+(2*a)^(1/3)*(a-b/x)^(1/3)+(2*a)^(1/3)*(2*a)^(1/3
))+59*((2*a)^(1/3))^2/96/sqrt(3)/a/b^4*atan(((a-b/x)^(1/3)+1/2*(2*a)^(1/3))/sqrt(3)*2/(2*a)^(1/3))+59*(2*a)^(1
/3)*(2*a)^(1/3)/288/b^4/a*ln(abs((a-b/x)^(1/3)-(2*a)^(1/3)))

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maple [F]  time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {a^{2} x^{2}+b^{2}}{\left (a^{2} x^{2}-b^{2}\right )^{3} \left (a \,x^{3}-b \,x^{2}\right )^{\frac {1}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2*x^2+b^2)/(a^2*x^2-b^2)^3/(a*x^3-b*x^2)^(1/3),x)

[Out]

int((a^2*x^2+b^2)/(a^2*x^2-b^2)^3/(a*x^3-b*x^2)^(1/3),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{2} x^{2} + b^{2}}{{\left (a^{2} x^{2} - b^{2}\right )}^{3} {\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*x^2+b^2)/(a^2*x^2-b^2)^3/(a*x^3-b*x^2)^(1/3),x, algorithm="maxima")

[Out]

integrate((a^2*x^2 + b^2)/((a^2*x^2 - b^2)^3*(a*x^3 - b*x^2)^(1/3)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int -\frac {a^2\,x^2+b^2}{{\left (b^2-a^2\,x^2\right )}^3\,{\left (a\,x^3-b\,x^2\right )}^{1/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(b^2 + a^2*x^2)/((b^2 - a^2*x^2)^3*(a*x^3 - b*x^2)^(1/3)),x)

[Out]

int(-(b^2 + a^2*x^2)/((b^2 - a^2*x^2)^3*(a*x^3 - b*x^2)^(1/3)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{2} x^{2} + b^{2}}{\sqrt [3]{x^{2} \left (a x - b\right )} \left (a x - b\right )^{3} \left (a x + b\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a**2*x**2+b**2)/(a**2*x**2-b**2)**3/(a*x**3-b*x**2)**(1/3),x)

[Out]

Integral((a**2*x**2 + b**2)/((x**2*(a*x - b))**(1/3)*(a*x - b)**3*(a*x + b)**3), x)

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